Polydivisible number

In mathematics a polydivisible number is a number in a given number base with digits abcde... that has the following properties:

Its first digit a is not 0.
The number formed by its first two digits ab is a multiple of 2.
The number formed by its first three digits abc is a multiple of 3.
The number formed by its first four digits abcd is a multiple of 4.
etc.
Definition

Let be a positive integer, and let be the number of digits in n written in base b. The number n is a polydivisible number if for all,
; Example
For example, 10801 is a seven-digit polydivisible number in base 4, as

Enumeration

For any given base, there are only a finite number of polydivisible numbers.

Maximum polydivisible number

The following table lists maximum polydivisible numbers for some bases b, where represent digit values 10 to 35.

Base	Maximum polydivisible number	Number of base-b digits
2		2
3		6
4		7
5		10
10		25
12		28

Estimate for ''F_b''(''n'') and Σ(''b'')

Let be the number of digits. The function determines the number of polydivisible numbers that has digits in base, and the function is the total number of polydivisible numbers in base.
If is a polydivisible number in base with digits, then it can be extended to create a polydivisible number with digits if there is a number between and that is divisible by. If is less or equal to, then it is always possible to extend an digit polydivisible number to an -digit polydivisible number in this way, and indeed there may be more than one possible extension. If is greater than, it is not always possible to extend a polydivisible number in this way, and as becomes larger, the chances of being able to extend a given polydivisible number become smaller. On average, each polydivisible number with digits can be extended to a polydivisible number with digits in different ways. This leads to the following estimate for :
Summing over all values of n, this estimate suggests that the total number of polydivisible numbers will be approximately

Base		Est. of	Percent Error
2	2		59.7%
3	15		-15.1%
4	37		8.64%
5	127		−7.14%
10	20456		-3.09%

Specific bases

All numbers are represented in base, using A−Z to represent digit values 10 to 35.

Base 2

Base 3

Base 4

Base 5

The polydivisible numbers in base 5 are
The smallest base 5 polydivisible numbers with n digits are
The largest base 5 polydivisible numbers with n digits are
The number of base 5 polydivisible numbers with n digits are

Length n	F₅	Est. of F₅
1	4	4
2	10	10
3	17	17
4	21	21
5	21	21
6	21	17
7	13	12
8	10	8
9	6	4
10	4	2

Base 10

The polydivisible numbers in base 10 are
The smallest base 10 polydivisible numbers with n digits are
The largest base 10 polydivisible numbers with n digits are
The number of base 10 polydivisible numbers with n digits are

Length n	F₁₀	Est. of F₁₀
1	9	9
2	45	45
3	150	150
4	375	375
5	750	750
6	1200	1250
7	1713	1786
8	2227	2232
9	2492	2480
10	2492	2480
11	2225	2255
12	2041	1879
13	1575	1445
14	1132	1032
15	770	688
16	571	430
17	335	253
18	180	141
19	90	74
20	44	37
21	18	17
22	12	8
23	6	3
24	3	1
25	1	1

Programming example

The example below searches for polydivisible numbers in Python.

def find_polydivisible -> list:
"""Find polydivisible number."""
numbers =
previous =
new =
digits = 2
while not previous :
numbers.append
for n in previous:
for j in range:
number = n * base + j
if number % digits 0:
new.append
previous = new
new =
digits = digits + 1
return numbers